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If $G$ is a group and $a,x\in G$, then would we call $x^{-1}ax$ a conjugate of $x$ or a conjugate of $a$?

Sorry for such a short question, was just doing a problem and want to call this something so it is easier for me to refer to and I want to call it what seems to be the normal way to say it.

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Thank you guys so much for your answers! –  Starlight Apr 30 '13 at 0:51
    
I'd call it the conjugate of $a$ by $x^{-1}$. I'd reserve the term conjugate of $a$ by $x$ for $x a x^{-1}$. –  Michael Joyce Apr 30 '13 at 2:52

2 Answers 2

up vote 4 down vote accepted

We call $x^{-1}ax$ a conjugate of $a$.

You can say that you conjugated $a$ by $x$.

You should note, in general, $a$ is NOT conjugate to $x$, for arbitrary elements of $G$.

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$x^{-1}ax$ is the conjugate of $a$ by $x$. http://mathworld.wolfram.com/Conjugation.html

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